Explanation Given information: The mass of the package ( m ) is 50 kg . The velocity of the package ( v 1 ) is 2 m / s . The distance between the package and cable ( l ) is 8 m . The angle of spring ( θ ) is 20 ° . The spring constant ( k ) is 30 kN / m . The coefficient of kinetic friction between the package ( μ k ) is 0.15. The compression of the spring ( e 1 ) is 50 mm or 0 .05m . The acceleration due to gravity ( g ) is 9.81 m / s 2 . Calculation: Consider the position 1. Assume the position 1 be the initial position which 8 m from the end of spring when it is compressed of 50 mm by cable. Calculate the kinetic energy ( T 1 ) in the position 1 using the formula: T 1 = 1 2 m v 1 2 Substitute 50 kg for m and 2 m / s for v . T 1 = 1 2 × 50 × ( 2 ) 2 = 100 J Calculate the height ( h 1 ) from base to package using the relation: h 1 = l sin θ Substitute 8 m for l and 20 ° for θ . h 1 = ( 8 ) sin 20 ° = 2.7362 m Calculate the potential energy stored due to elongation at position 1 ( V 1 ) e using the formula: ( V 1 ) e = 1 2 k e 1 2 Substitute 0.050 m for e 1 and 30 kN / m for k . ( V 1 ) e = 1 2 × 30 × 10 3 × ( 0.050 ) 2 = 37.5 J Calculate the potential energy stored due to gravitation at position 1 ( V 1 ) g using the formula: ( V 1 ) g = m g h 1 Substitute 50 kg for m , 9.81 m / s 2 for g , and 2.7362 m for h 1 . ( V 1 ) g = 50 × 9.81 × ( 2.7362 m ) = 1342.1 J Calculate the potential energy stored in rod in position 1 ( V 1 ) using the relation: V 1 = ( V 1 ) e + ( V 1 ) g Substitute 37.5 J for ( V 1 ) e and 1342.1 J for ( V 1 ) g . V 1 = 37.5 J+1342 .1J = 1379.6 J Consider the position 2. Assume the position 2 be the maximum compression position. Calculate the kinetic energy ( T 2 ) in the position 2 using the formula: T 2 = 1 2 m v 2 2 Substitute 50 kg for m and 0 for v . T 2 = 1 2 × 50 × ( 0 ) 2 = 0 Calculate the height ( h 2 ) using the relation: h 2 = − x sin θ Here, x is the addition maximum deformation of the spring. Substitute 20 ° for θ . h 2 = − x sin 20 ° = − 0.34202 x Calculate the compression of the spring e 2 in position 2 using the relation: e 2 = e 1 + x Substitute 0.050 m for e 1 . e 2 = 0.050 + x Calculate the potential energy stored due to elongation at position 2 ( V 2 ) e using the formula: ( V 2 ) e = 1 2 k e 2 2 Substitute 0.050 + x for e 2 and 30 kN / m for k

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 13.2, Problem 13.69P

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Find the maximum additional deformation of the spring (x) in bringing the package to rest.

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Find the maximum additional deformation of the spring ( x ) in bringing the package to rest.

Solution Summary: The author calculates the maximum additional deformation of the spring in bringing the package to rest using the formula: underset_0.2012m

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Find the maximum additional deformation of the spring (x) in bringing the package to rest.

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